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Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game

Author

Listed:
  • Arvind Srinivasan

    (University of Colorado Denver)

  • Burton Simon

    (University of Colorado Denver)

Abstract

The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, $$(p(1),p(2),\ldots )$$ ( p ( 1 ) , p ( 2 ) , … ) , where p(k) is the probability you pick k. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, n, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large n, players should choose a number uniformly between 1 and $$\phi _n \sim O(n/\ln (n))$$ ϕ n ∼ O ( n / ln ( n ) ) . Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for $$\phi _n$$ ϕ n , and efficient ways to sample from the continuous approximation.

Suggested Citation

  • Arvind Srinivasan & Burton Simon, 2024. "Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 653-671, June.
  • Handle: RePEc:spr:jogath:v:53:y:2024:i:2:d:10.1007_s00182-023-00881-0
    DOI: 10.1007/s00182-023-00881-0
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    References listed on IDEAS

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    1. Simone Pigolotti & Sebastian Bernhardsson & Jeppe Juul & Gorm Galster & Pierpaolo Vivo, 2011. "Equilibrium strategy and population-size effects in lowest unique bid auctions," Papers 1105.0819, arXiv.org, revised Feb 2012.
    2. Roger B. Myerson, 1998. "Population uncertainty and Poisson games," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 375-392.
    3. Simone Pigolotti & Sebastian Bernhardsson & Jeppe Juul & Gorm Galster & Pierpaolo Vivo, 2012. "Equilibrium strategy and population-size effects in lowest unique bid auctions," Post-Print hal-00681002, HAL.
    4. Robert Östling & Joseph Tao-yi Wang & Eileen Y. Chou & Colin F. Camerer, 2011. "Testing Game Theory in the Field: Swedish LUPI Lottery Games," American Economic Journal: Microeconomics, American Economic Association, vol. 3(3), pages 1-33, August.
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